I showed my current employer my cert and they didn't know what it was. I've never been asked for that either.

My employer said I didn't have a 'TEFL', when I replied I had a CELTA he just said 'but that's not a TEFL'...... Yet a non-native English teacher, who did an online cert that had the word 'master' in it, goes around proclaiming they've got a master's degree and is treated as such... Welcome to China.

Can you please tell me which of these are not required by China, if any?
If I'm missing any for China, can you tell me what they are?

Presumably you mean Korea. China is hugely different in that in general Korea applies the laws and requirements regarding visas across the country. China doesn't do this kind of thing uniformly and hence different rules apply depending on province, particular PSB office, the PSB officer your school deals with, guangxi your potential employer has with the powers that be, how willing your boss is to bend the rules a little, day of the week, the weather outside etc.
Some schools and provinces require reference letters confirming 2 years of previous employment, others don't. Some need TEFL certs, others don't. There's rumors of some provinces now requiring criminal checks from home (Henan for example), medical checks needed in advance, other times not.
Get the basics and confirm the exact paperwork required when you have a decent school lined up.
Apostilles are needed (yet), thank god.

I showed my current employer my cert and they didn't know what it was. I've never been asked for that either.

My employer said I didn't have a 'TEFL', when I replied I had a CELTA he just said 'but that's not a TEFL'...... Yet a non-native English teacher, who did an online cert that had the word 'master' in it, goes around proclaiming they've got a master's degree and is treated as such... Welcome to China.

Maybe I should show my certificate that says "ASE Master Heavy Duty Truck Technician" and then they will think I also have a "Masters Degree?"